Chapter 8.7, Problem 14E

In forming the Koch snowflake in Figure 8.104 on page 510, the perimeter becomes greater at each step in the process. If each side of the original triangle is 1 unit, a general formula for the perimeter, L, of the snowflake at any step, n, may be found by the formula

$\text{L}=3{\left(\frac{4}{3}\right)}^{\text{n }-1}$

For example, at the first step when n = 1, the perimeter is 3 units, which can be verified by the formula as follows:

$\text{L}=3{\left(\frac{4}{3}\right)}^{1-1}=3{\left(\frac{4}{3}\right)}^0=3\cdot 1=3$

At the second step, when n = 2, we find the perimeter as follows:

$\text{L}=3{\left(\frac{4}{3}\right)}^{2-1}=3\left(\frac{4}{3}\right)=4$

Thus, at the second step the perimeter of the snowflake is 4 units.

a. Use the formula to complete the following table.

b. Use the results of your calculations to explain why the perimeter of the Koch snowflake is infinite.

c. Explain how the Koch snowflake can have an infinite perimeter, but a finite area.